Let \(k= k(\eta )\)  be the function field of \(S\) . Then, for all \(s\)  in \(B\) , we have \(\operatorname{End}_{\operatorname{Gal}(\overline{k}/k)}(T_\ell \mathcal {A}_\eta ) \otimes \mathbb {Q}_\ell \cong \operatorname{End}_{\operatorname{Gal}(\overline{k(s)}/k(s))}(T_\ell \mathcal {A}_s) \otimes \mathbb {Q}_\ell \) . By Tate's conjecture on endomorphisms (which holds for all finitely generated fields of characteristic 0, see [1]}) we thus obtain \(\operatorname{End}_{k}( \mathcal {A}_\eta ) \otimes \mathbb {Q}_\ell \cong \operatorname{End}_{k(s)}(\mathcal {A}_s) \otimes \mathbb {Q}_\ell \) . Since the inclusion \(\operatorname{End}_{k}( \mathcal {A}_\eta ) \hookrightarrow \operatorname{End}_{k(s)}(\mathcal {A}_s)\)  has torsion-free cokernel, we see that all points in \(B\)  satisfy the desired property concerning endomorphisms and have bounded degree over \(\mathbb {Q}\)  (we take in particular \(d_0 = \deg \pi \) ). Note that \(B\)  is Zariski dense, so that \(\cup _{h\ge 1} S(d_0,h)_{\mathrm {End}}\)  is Zariski dense. To conclude the proof, it only remains to show that the cardinality of \(S(d_0,h)_{\mathrm {End}}\)  grows exponentially in \(h\) . This follows from the fact that the complement of a thin set of \(\mathbb {A}_{\mathbb {Q}}^{\dim S}\)  contains \(\gg e^{ch}\)  points of logarithmic height \(h\)  for some positive constant \(c\)  (see [2]}).
